representation and character theory in 2-categories · 2017-02-26 · for the character of the...

Advances in Mathematics 217 (2008) 2268–2300www.elsevier.com/locate/aim

Representation and character theory in 2-categories ✩

Nora Ganter a, Mikhail Kapranov b,∗

a Department of Mathematics, Colby College, Mayflower Hill 5830, Waterville, ME 04901, USAb Yale University, Department of Mathematics, 10 Hillhouse Avenue, New Haven, CT, USA

Received 12 January 2007; accepted 17 October 2007

Available online 22 January 2008

Communicated by Michael J. Hopkins

Abstract

We develop a (2-)categorical generalization of the theory of group representations and characters. Wecategorify the concept of the trace of a linear transformation, associating to any endofunctor of any smallcategory a set called its categorical trace. In a linear situation, the categorical trace is a vector space andwe associate to any two commuting self-equivalences a number called their joint trace. For a group actingon a linear category V we define an analog of the character which is the function on commuting pairs ofgroup elements given by the joint traces of the corresponding functors. We call this function the 2-characterof V . Such functions of commuting pairs (and more generally, n-tuples) appear in the work of Hopkins,Kuhn and Ravenel [Michael J. Hopkins, Nicholas J. Kuhn, Douglas C. Ravenel, Generalized group charac-ters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (3) (2000) 553–594 (electronic)]on equivariant Morava E-theories. We define the concept of induced categorical representation and showthat the corresponding 2-character is given by the same formula as was obtained in [Michael J. Hopkins,Nicholas J. Kuhn, Douglas C. Ravenel, Generalized group characters and complex oriented cohomologytheories, J. Amer. Math. Soc. 13 (3) (2000) 553–594 (electronic)] for the transfer map in the second equivari-ant Morava E-theory.© 2007 Elsevier Inc. All rights reserved.

Keywords: 2-categories; Character; 2-vector spaces

✩ Ganter was supported by NSF grant DMS-0504539. Kapranov was supported by NSF grant DMS-0500565. Thepaper was completed while Ganter was visiting MSRI.

* Corresponding author.E-mail address: [email protected] (M. Kapranov).

0001-8708/$ – see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.aim.2007.10.004

N. Ganter, M. Kapranov / Advances in Mathematics 217 (2008) 2268–2300 2269

1. Introduction

The goal of this paper is to develop a (2-)categorical generalization of the theory of group rep-resentations and characters. It is classical that a representation � of a group G is often determinedby its character

χ(g) = tr(�(g)

),

which is a class function on G.Remarkably, generalizations of character theory turn up naturally in the context of homotopy

theory. Since this so-called Hopkins–Kuhn–Ravenel character theory motivated much of ourwork, we will start with a short and very informal summary of it and some other homotopytheoretic topics. Fix a prime p, let n be a natural number, and let BG denote the classifyingspace of G. Assume that G is finite. In [21], Hopkins, Kuhn and Ravenel computed the ringE∗

n(BG), where En is a generalized cohomology theory (depending on p), which was introducedby Morava [36]. The first Morava E-theory is p-completed K-theory,

E1 = Kp.

Hopkins, Kuhn and Ravenel found that elements χ ∈ E∗n(BG) are most naturally described as

n-class functions, i.e. functions

χ(g1, . . . , gn)

defined on n-tuples of commuting elements of G and invariant under simultaneous conjugation.In the context of [21], all the gi are required to have p-power order. Hopkins, Kuhn and Ravenelactually make a much stronger case that the n-class functions occurring in this way should beviewed as generalized group characters: if α : H ↪→ G is the inclusion of a subgroup, then thereis a map

Bα : BH → BG,

and in the stable homotopy category, one has a transfer map τα in the other direction. Thesemaps make the correspondence G �→ En(BG) into a Mackey functor. The map E∗

n(τα) sendsa generalized character of H to a generalized character of G. If we stick with the analogy toclassical character theory, it plays the role of the induced representation. Hopkins, Kuhn andRavenel compute its effect on generalized characters. They find that it is described by the formula

E∗n(τα)(χ)(g1, . . . , gn) = 1

|H |∑s∈G|

s−1gs∈Hn

χ(s−1g1s, . . . , s

−1gns), (1)

where g = (g1, . . . , gn) is an n-tuple of commuting elements in G. For n = 1, this is the formulafor the character of the induced representation, cf. [38].

The number n is often referred to as the chromatic level of the theory, see [35] for generalbackground on the chromatic picture in homotopy theory. In the case n = 2, the theory E2 is anexample of an elliptic cohomology theory. For background on elliptic cohomology, we refer thereader to the introduction of [1].

2270 N. Ganter, M. Kapranov / Advances in Mathematics 217 (2008) 2268–2300

Just as the representation ring R(G) may be viewed as equivariant K-theory of the one pointspace, the ring E2(BG) is Borel equivariant E2-theory of the one point space. Elliptic coho-mology is a field at the intersection of several areas of mathematics, and there is a variety offields that have motivated definitions of equivariant elliptic cohomology. To name a few, wehave Devoto’s definition, motivated by orbifold string theory [11], we have Grojnowski’s work[19], motivated by the theory of loop groups, we have the axiomatic approach in [18], involvingprincipal bundles over elliptic curves, we have a connection to generalized Moonshine (cf. [11,15], and [4]), and we have recent constructions of Lurie and Gepner [17,28], which satisfyaxioms similar to those of [18] but formulated in the context of derived algebraic geometry.Lurie’s construction naturally involves 2-groups. It is remarkable that each of these construc-tions, in one way or another, leads to class-functions on pairs of commuting elements of thegroup.

What is lacking in the above approaches is an analog of the notion of representation whichwould produce the generalized characters by means of some kind of trace construction. In thispaper, we supply such a notion (for n = 2). It turns out that the right object to consider is anaction of G on a category instead of a vector space or, more generally, on an object of a 2-category.

Our main construction is the so-called categorical trace of a functor A : V → V from a smallcategory V to itself (or, more generally, of a 1-endomorphism of an object of a 2-category).This categorical trace is a set, denoted Tr(A), see Definition 3.1. If k is a field, and the cat-egory V is k-linear, then Tr(A) is a k-vector space. In the latter case, given two commutingself-equivalences A,B : V → V , we define their joint trace τ(A,B) to be the ordinary trace ofthe linear transformation induced by B on the vector space Tr(A). Here we assume that Tr(A)

is finite-dimensional.For a group G acting on V this gives a 2-class function called the 2-character of the cate-

gorical representation V . Among other things, we define an induction procedure for categoricalrepresentations and show that it produces the map (1) on the level of characters.

This very simple and natural construction ties in with other geometric approaches to ellipticcohomology. Already Segal, in his Bourbaki talk [37], proposed to look for some kind of “el-liptic objects” which are related to vector bundles in the same way as 2-categories are related toordinary categories. While vector bundles can be equipped with connections and thus with theconcept of parallel transport along paths, one expects elliptic objects to allow parallel transportalong 2-dimensional membranes. Similarly, more recent works [3,22,40] aiming at geometricdefinitions of elliptic cohomology, all involve 2-categorical constructions.

The present paper is kept at an elementary level and does not require any knowledge of homo-topy theory (except for the final Section 8 devoted to the comparison with [21]). Nevertheless,the connection with homotopy theory and, in particular, with equivariant elliptic cohomologywas important for us. It provided us with a motivation as well as with a possible future field ofapplications.

We also do not attempt to discuss actions of groups on n-categories for n > 1 which seem tobe the right way to get (n + 1)-class functions.

We have recently learned of a work in progress by Bruce Bartlett and Simon Willerton [5]where, interestingly, the concept of the categorical trace also appears although the motivation isdifferent.


2. Background on 2-categories

2.1. Recall [31] that a 2-category C consists of the following data:

(1) A class of objects ObC.(2) For any x, y ∈ ObC a category HomC(x, y). Its objects are called 1-morphisms from x to y

(notation A : x → y). We will also use the notation 1 HomC(x, y) for Ob HomC(x, y). Forany A,B ∈ 1 HomC(x, y) morphisms from A to B in HomC(x, y) are called 2-morphismsfrom A to B (notation φ : A ⇒ B). We denote the set of such morphisms by

2 HomC(A,B) = HomHomC(x,y)(A,B).

The composition in the category HomC(x, y) will be denoted by ◦1 and called the verticalcomposition. Thus if φ : A ⇒ B and ψ : B ⇒ C, then ψ ◦1 φ : A ⇒ C.

(3) The composition bifunctor

HomC(y, z) × HomC(x, y) → HomC(x, z)

(A,B) �→ A ◦0 B.

In particular, for a pair of 1-morphisms A,B : x → y, a 2-morphisms φ : A ⇒ B betweenthem, and a pair of 1-morphisms C,D : y → z and a 2-morphism ψ : C ⇒ D between them,there is a composition ψ ◦0 φ : C ◦0 A ⇒ D ◦0 B .

(4) The natural associativity isomorphism

αA,B,C : (A ◦0 B) ◦0 C ⇒ A ◦0 (B ◦0 C).

It is given for any three composable 1-morphisms A,B,C and satisfies the pentagonal ax-iom, see [31].

(5) For any x ∈ ObC, a 1-morphism 1x ∈ 1 HomC(x, x) called the unit morphism, which comesequipped with 2-isomorphisms

εφ : 1x ◦0 A ⇒ A, for any A : y → x,

ζψ : B ◦ 1x ⇒ B, for any B : x → z,

satisfying the axioms of [31].

We denote by 1 MorC and 2 MorC the classes of all 1- and 2-morphisms of C. We say that Cis strict if all the αA,B,C , εφ and ζψ are identities (in particular the source and target of each ofthem are equal). It is a theorem of Mac Lane and Paré that every 2-category can be replaced bya (2-equivalent) strict one. See [31] for details.

2.2. Examples

(a) The 2-category Cat has, as objects, all small categories, as morphisms their functors andas 2-morphisms natural transformations of functors. This 2-category is strict. We will use thenotation Fun(A,B) for the set of functors between categories A and B (i.e., 1-morphisms in


Cat) and NT(F,Φ) for the set of natural transformations between functors F and Φ . Most of theexamples of 2-categories can be embedded into Cat: a 2-category C can be realized as consistingof categories with some extra structure.

(b) Let k be a field. The 2-category 2 Vectk , see [26] has, as objects, symbols [n], n =0,1,2, . . . . For any two such objects [m], [n] the category Hom([m], [n]) has, as objects, 2-matrices of size m by n, i.e., matrices of vector spaces A = ‖Aij‖, i = 1, . . . ,m, j = 1, . . . , n.Morphisms between 2-matrices A and B of the same size are collections of linear mapsφ = {φij :Aij → Bij }. Composition of 2-matrices is given by the formula

(A ◦ B)ij =⊕

l

Ail ⊗ Blj .

This 2-category is not strict. An explicit strict replacement was constructed in [12].(c) The 2-category Bim has, as objects, associative rings. If R,S are two such rings, then

HomBim(R,S) is the category of (R,S)-bimodules. The composition bifunctor

Hom(S,T ) × Hom(R,S) → Hom(R,T )

is given by the tensor product:

(M,N) �→ N ⊗S M.

This 2-category is also not strict.Relation to Cat: To a ring R we associate the category Mod -R of right R-modules. Then each

(R,S)-bimodule M defines a functor

Mod -R → Mod -S, P �→ P ⊗R M.

The 2-category 2 Vectk is realized inside Bim by associating to [m] the ring k⊕m. An m by n

2-matrix is the same as a (k⊕m,k⊕n)-bimodule.We will denote by Bimk the sub-2-category in Bim formed by k-algebras as objects and the

same 1- and 2-morphisms as in Bim.(d) Let X be a CW-complex. The Poincaré 2-category Π(X) has, as objects, points of X,

as 1-morphisms Moore paths [0, t] → X and as 2-morphisms homotopy classes of homotopiesbetween Moore paths.

We will occasionally use the concept of a (strong) 2-functor Φ : C →D between 2-categories C and D. Such a 2-functor consists of maps ObC → ObD, 1 MorC → 1 MorD and2 MorC → 2 MorD preserving the composition of 2-morphisms and preserving the compositionof 1-morphisms up to natural 2-isomorphisms. See [31] for details.

2.3. 2-categories with extra structure

We recall the definition of an enriched category from [31] or [24]. Let (A,⊗, S) be a closedsymmetric monoidal category, so ⊗ is the monoidal operation and S is a unit object.


Definition 2.1. A category enriched over A (or simply an A-category) C is defined in the sameway as a category, with the morphism sets replaced by objects hom(X,Y ) of A and compositionreplaced by A-morphisms

hom(X,Y ) ⊗ hom(Y,Z) → hom(X,Z),

with units

1X : S → hom(X,X),

such that the usual associativity and unit diagrams commute.

Example 2.2. Categories enriched over the category of abelian groups are commonly known aspre-additive categories. By an additive category one means a pre-additive category possessingfinite direct sums. If k is a field, categories enriched over the category of k-vector spaces areknown as k-linear categories.

Example 2.3. A strict 2-category is the same as a category enriched over the category of smallcategories with ⊗ being the direct product of categories (cf. [39]).

Definition 2.4. Let A be a category. A strict 2-category C enriched over A, or shorter an A-2-category, is a category enriched over the category of small A-categories.

Definition 2.5. We define a strict pre-additive 2-category to be a 2-category enriched over thecategory of abelian groups. Let k be a field. Then a (strict) k-linear 2-category is defined to be a2-category enriched over the category of k-vector spaces. Weak additive and k-linear 2-categoriesare defined in a similar way.

We will freely use the concept of a triangulated category [16,33]. If D is triangulated, then wedenote by X[i] the i-fold iterated shift (suspension) of an object X in D. We will denote

Hom•D(X,Y ) =

⊕i

HomD(X,Y [i]).

We will call a 2-category C triangular if each HomC(x, y) is made into a triangulated categoryand the composition functor is exact in each variable.

2.4. Examples

(a) The 2-category Bim is additive. The 2-categories 2 Vectk and Bimk are k-linear.(b) Define the 2-category DBim to have the same objects as Bim, i.e., associative rings. The

category HomDBim(R,S) is defined to be the derived category of complexes of (R,S)-bimodulesbounded above. The composition is given by the derived tensor product:

(M,N) �→ N ⊗LS M.

This gives a triangular 2-category.


(c) The 2-category Vark has as objects smooth projective algebraic varieties over k. If X,Y

are two such varieties, then

HomVark(X,Y ) = DbCoh(X × Y)

is the bounded derived category of coherent sheaves on X × Y . If K ∈ DbCoh(Y × Z) andL ∈ DbCoh(X × Y), then their composition is defined by the derived convolution

K ∗L= Rp13∗(p∗

12L⊗L p∗23K

),

where p12,p13,p23 are the projections of X × Y × Z to the products of two factors. This againgives a triangular 2-category.

Relation to Cat: To every variety X we associate the category DbCoh(X). Then every sheafK ∈ DbCoh(X × Y) (“kernel”) defines a functor

FK : DbCoh(X) → DbCoh(Y ), F �→ Rp2∗(p∗

1F ⊗L K),

and FK∗L is naturally isomorphic to FK ◦FL. It is not known, however, whether the natural map

HomDbCoh(X,Y )(K,L) → NT(FK,FL)

is a bijection for arbitrary K,L. So in practice the source of this map is used as a substitute forits target.

(d) The 2-category RAnk has, as objects, real analytic manifolds. For any two such mani-folds X,Y the category HomCW (X,Y ) is defined to be DbConstr(X × Y), the bounded derivedcategory of (R-)constructible sheaves of k-vector spaces on X × Y , see [25, Section 8.4], forbackground on constructible sheaves. The composition is defined similarly to the above, withp∗

ij understood as sheaf-theoretic direct images rather than O-module-theoretic direct images.This is a triangular 2-category.

Relation to Cat: To every real analytic manifold X we associate the category DbConstr(X).Then, as in (c), any sheaf

K ∈ DbConstr(X × Y)

can be considered as a “kernel” defining a functor

DbConstr(X) → DbConstr(Y ).

(e) Let Ab denote the category of all abelian categories. For any such categories A,B thecategory Fun(A,B) is again abelian: a sequence of functors is exact if it takes any objectinto an exact sequence. So we have a triangular 2-category DAb with same objects as Ab butHom(A,B) = DbFun(A,B).


3. The categorical trace

3.1. The main definition

As motivation consider the 2-category 2 Vectk . In this situation there is a naïve way to definethe “trace” of a 1-automorphism, namely as direct sum of the diagonal entries of the matrix.This naïve notion of trace is equivalent to the following definition that makes sense in any2-category C:

Definition 3.1. Let C be a 2-category, x an object of C and A : x → x a 1-endomorphism of x.The categorical trace of A is defined as

Tr(A) = 2 HomC(1x,A).

If C is triangular, we write

Tri (A) = Tr(A[i]), i ∈ Z, Tr•(A) =

⊕i

Tri (A).

Remark 3.2 (Functoriality). Note that for each x, the categorical trace defines a functor

Tr : 1 End(x) → Set

φ ∈ 2 Hom(A,B) �→ φ∗,

where

φ∗ : Tr(A) → Tr(B)

is given by composition with φ. A priori, Tr is set-valued, but if we assume C to be enriched overa category A (cf. Definition 2.4), Tr takes values in A. We will often assume that C is k-linearfor a fixed field k.

3.2. Examples of the categorical trace

Our first example is the motivational example mentioned above.

Example 3.3 (2-vector spaces). Let C = 2- Vectk and x = [n]. Then A is an n × n matrix A =(Aij ), where the Aij are vector spaces. In this case,

Tr(A) =n⊕

i=1

Aii.

Example 3.4 (Categories). Let C = Cat and x = V a category, so A : V → V is an endofunctor.Then Tr(A) = NT(idV ,A) is the set of natural transformations from the identity functor to A.


Example 3.5 (Bimodules). Let C = DBim, so that x = R is a ring, and A = M is an R-bimodule.Then

Tr•(A) = Ext•R⊗Rop(R,M)

is the Hochschild cohomology of R with coefficients in M , see [27].

Example 3.6 (Varieties). Let C = Vark , x = X be a variety, and A = K be a complex of coherentsheaves on X × X. Then

Tr•(A) = H•(X, i!(K)

).

Here i : X → X × X is the diagonal embedding, i! is the right adjoint of i∗, and H is the hyper-cohomology. In particular, if K is a vector bundle on X × X situated in degree 0, then

Tr•(A) = H •(X,K|Δ)

is the cohomology of the restriction of K to the diagonal.

3.3. The center of an object

The set Tr(1x) will be called the center of x and denoted Z(x). It is closed under both com-positions ◦0 and ◦1. The following fact is well known [31].

Proposition 3.7. The operations ◦0 and ◦1 on Z(x) coincide and make it into a commutativemonoid.

Thus, if C is pre-additive, then Z(x) is a commutative ring and for each A : x → x the groupTr(A) is a Z(x)-module.

3.4. Examples

(a) If C = Ab and x = V is an abelian category, then Z(V), i.e., the ring of natural transfor-mations from the identity functor to itself is known as the Bernstein center of V , see [7].

(b) If C = Π(X) is the Poincaré 2-category of a CW-complex X, then Z(x) = π2(X,x) is thesecond homotopy group. Proposition 3.7 is the categorical analog of the commutativity of π2.

3.5. Conjugation invariance of the categorical trace

In this section we assume for simplicity that the 2-category C is strict. Recall [31] that a1-morphism B : y → x is called an equivalence if there exist a 1-morphism C : x → y calledquasi-inverse and 2-isomorphisms u : 1x ⇒ BC, v : 1y ⇒ CB . For any object x, the 1-morphismB = 1x is an equivalence with C = 1x and u,v the isomorphisms from 2.1(4). If B and B ′ arecomposable 1-morphisms, which are equivalences with quasi-inverses (C,u, v) and (C′, u′, v′)respectively, then B ′′ ◦0 B is an equivalence with quasi-inverse(

C ◦0 C′, (B ′ ◦0 u ◦0 C′) ◦1 u′, (C ◦0 v′ ◦0 B) ◦1 v). (2)


Proposition 3.8 (Conjugation invariance).

(a) Let A : x → x be a 1-endomorphism and B : x → y an equivalence with quasi-inverse C.Then the rule

(φ : 1x ⇒ A) �→ (B ◦0 φ ◦0 C) ◦1 u

defines a bijection of sets

ψ(B,C,u, v) : Tr(A) → Tr(BAC).

By abuse of notation, we will write ψ(B) when C, u and v are clear from the context.(b) Assume that B and B ′ are 1-endomorphisms of x and that both of them are equivalences.

Then we have

ψ(B ′ ◦0 B) = ψ(B ′) ◦ ψ(B).

(c) We have ψ(1x) = id.

Proof. (a) To explain the formula, note that we can view u as a 2-morphism 1y ⇒ BC =B ◦ 1x ◦ C, while

B ◦0 φ ◦0 C : B ◦ 1x ◦ C ⇒ B ◦ A ◦ C.

Since u is a 2-isomorphism, composing with u is a bijection.(b) This follows from the definition of ψ(B) together with (2).(c) Obvious. �

Proposition 3.9. Let C be an additive 2-category and A,A′ : x → x be two 1-morphisms. Then

Tr(A ⊕ A′) = Tr(A) ⊕ Tr(A′).

This is an immediate consequence of the fact that H = HomC(x, x) is an additive category,and that therefore

HomH(1x,A ⊕ A′) = HomH(1x,A) ⊕ HomH(1x,A′).

3.6. The joint trace

In the situation of Proposition 3.5 assume that A and B commute, i.e., that we are given a2-isomorphism

η : B ◦ A ⇒ A ◦ B.

Then we have a map

B∗ : Tr(A) → Tr(A),


defined as the composition

Tr(A)ψ(B)−−−→ Tr(BAC)

Tr(η◦01)−−−−−→ Tr(ABC)Tr(1◦0u

−1)−−−−−−→ Tr(A).

Assume now that the 2-category C is k-linear for a field k. Then Tr(A) is a k-vector space, andB∗ is a linear operator. Let us further assume that Tr(A) is finite-dimensional. Then we definethe joint trace of A and B to be the following element of k:

τ(A,B) = Trace{B∗ : Tr(A) → Tr(A)

}.

It depends on the choice of the commutativity isomorphism η, as well as on the equivalence datafor Φ .

4. 2-representations and their characters

4.1. 2-representations

Let G be a group. We view G as a 2-category with one object, pt, the set of 1-morphismsHom(pt,pt) = G and all the 2-morphisms being the identities of the above 1-morphisms.

Definition 4.1. Let C be a 2-category. A 2-representation of G in C is a strong 2-functor from G

to C. More explicitly, this is a system of the following data:

(a) an object V of C,(b) for each element g ∈ G, a 1-automorphism ρ(g) of V ,(c) for any pair of elements (g,h) of G a 2-isomorphism

φg,h : (ρ(g) ◦ ρ(h)) ∼= ⇒ρ(gh),

(d) and a 2-isomorphism

φ1 : ρ(1)∼= ⇒ idc,

such that the following conditions hold

(e) for any g,h, k ∈ G we have

φ(gh,k)

(φg,h ◦ ρ(k)

) = φ(g,hk)

(ρ(g) ◦ φh,k

)(associativity); we also write φg,h,k ,

(f) we have

φ1,g = φ1 ◦ ρ(g) and φg,1 = ρ(g) ◦ φ1.

Note that this definition is the special case of the concept of a representation of a 2-group asdefined by Elgueta [13, Def. 4.1]. This case corresponds to the 2-group being discrete, i.e., beingreduced to an ordinary group. Compare also [10, §0]. If D and C are 2-categories, then strong


2-functors from D to C form a 2-category Hom(D,C), see [20, Def. I.1.9] for strict 2-categoriesor [6] for the general case. In particular, 2-representations of G in C form a 2-category. We willdenote it by 2 RepC(G). We understand that the implications of this fact will be spelled out indetail in [5].

4.2. The category of equivariant objects

Consider the particular case of Definition 4.1 when C = Cat is the 2-category of (small) cate-gories. Then a 2-representation of G in C is the same as an action of G on a category V . In otherwords, each

ρ(g) : V → V

is a functor and each φg,h is a natural transformation. We will call a category with a G-actiona categorical representation of G and will denote by 2 Rep(G) = 2 RepCat(G) the 2-categoryformed by categorical representations.

In this section, we formulate the categorical analogue the concept of the subspace of G-invariants of a representation.

Definition 4.2. Fix a category 1 with one object and one morphism. The trivial 2-representationof G is given by the unique action of G on 1. We will also denote it by 1. Let ρ be an action ofG on V . We define the category of G-equivariant objects in V , denoted VG, to be the categoryof G-functors from 1 to ρ:

VG = Hom2 Rep(G)(1, ρ).

This definition spells out to the following. An object of VG consists of an object X ∈ Ob(V)

and a system

(εg : X → ρ(g)(X), g ∈ G

),

where εg are isomorphisms satisfying the following compatibility conditions: First, it is requiredthat for g = 1 we have

ε1 = φ−11,X : X �→ ρ(1)(X).

Second, it is required that for any g,h ∈ G the diagram

Xεg

εgh

ρ(g)(X)

ρ(g)(εh)

ρ(gh)(X) ρ(g)(ρ(h)(X))φg,h,X

is commutative.


Example 4.3. Let W be a category. Define the trivial action of G on W by taking all ρ(g) andφg,h to be the identities. Then a G-equivariant object in W is the same as a representation of G

in W , i.e., an object X ∈W and a homomorphism G → AutW (X).

Proposition 4.4. Let W be a category equipped with trivial G-action as in Example 4.3. Thenwe have an equivalence of categories

Hom2 Rep(G)(W,V) � HomCat(W,VG

).

In particular, taking W = pt (the category with one object and one morphism), we get

VG � Hom2 Rep(G)(pt,V).

In plain words, this means that any G-functor from W to V factors through the forgetfulfunctor

iV : VG → V .

In 2-categorical terms, this can be formulated by saying that the 2-functor

IG : 2 Rep(G) → Cat, V �→ VG,

is right 2-adjoint (in the sense of [20, Def. I.1.10]), to the 2-functor Cat → 2 Rep(G) associatingto any W the same category W with trivial G-action.

Proof. This follows at once from the definition of the Hom-categories in 2 Rep(G), which areparticular cases of Hom-categories in 2-categories of 2-functors, see [20, Definition I.1.9]. In-deed, denote by ρ the trivial action of G on W . Then a G-functor F : W → V gives, for eachobject X ∈ W , an object F(X) ∈ W together with isomorphisms

ug,X : F (ρ(g)(X)

) → ρ(g)(F(X)

),

satisfying the compatibility condition for each pair g,h ∈ G. Since ρ(g)(X) = X, the systemformed by F(X) and the ug,X gives an equivariant object of V . We leave further details to thereader. �Remark 4.5. The concept of the category of equivariant objects relates our approach to 2-representations with a different approach due to Ostrik [34]. If k is a field, then finite-dimensionallinear representations of G over k form a monoidal category (Rep(G),⊗) with respect to theusual tensor product. In [34] it was proposed to study module categories over Rep(G). In oursituation, given a G-action on a k-linear additive category V , the category VG is naturally amodule category over Rep(G). In other words, the tensor product of a G-representation and aG-equivariant object is again a G-equivariant object. It seems that in general, the passage from aG-category V to the Rep(G)-module category VG leads to some loss of information. However,in some particular cases, the two approaches are equivalent, see Remark 7.4 below.


4.3. Characters of 2-representations

We are now ready to define the categorical character of a 2-representation. To motivate thediscussion of this section, we start with a reminder of classical character theory.

4.3.1. Group characters and class functionsWe fix a field k of characteristic 0 containing all roots of unity. Let G be a group. Recall that

a function f : G → k is called a class function if it is invariant under conjugation.

Notation 4.6. We denote by Cl(G; k) the ring of class functions on G. As before, let Rep(G)

be the category of finite-dimensional representations of G over k. We write R(G) for itsGrothendieck ring K(Rep(G)).

If ρ : G → Aut(V ) is a representation, then its character

χV : G → k

g �→ Trace(ρ(g)

)is a class function. The following is well known [38].

Proposition 4.7. If G is finite, then the correspondence V �→ χV induces an isomorphism ofrings

R(G) ⊗ k → Cl(G; k).

4.3.2. The categorical characterThe classical definitions discussed in the previous section suggest the following analogues for

2-representations:

Definition 4.8. Let ρ be a 2-representation of G. We define the categorical character of ρ to bethe assignment

g �→ Tr(ρ(g)

).

We now discuss the sense in which the categorical character is a class function. First we recallthe definition of the inertia groupoid of G:

Definition 4.9. Let G be a group. The inertia groupoid Λ(G) of G is the category that has asobjects the elements of G and

HomΛ(G)(u, v) = {g ∈ G: v = gug−1}.

Proposition 4.10. Let C be a 2-category and let ρ be a 2-representation of G in C. Then thecategorical character of ρ is a functor from the inertia groupoid Λ(G) to the category of sets:

Tr(ρ) : Λ(G) → Set.


If C is enriched over a category A, then this functor takes values in A. In other words, for anyf,g ∈ G there is an isomorphism

ψ(g) = ψf (g) : Tr(ρ(f )

) → Tr(ρ(gfg−1

)),

and these isomorphisms satisfy

(a) ψ(gh) = ψ(g) ◦ ψ(h) and(b) ψ(1) = idρ(f ).

Proof. Pick f,g ∈ G and write A for ρ(f ), B for ρ(g), C for ρ(g−1) and define

u : 1c → BC

as the composite of maps from Definition 4.1:

u := φ−1g,g−1φ

−11 .

With this notation, Proposition 3.8(a) implies the existence of an isomorphism

ψ ′ : Tr(ρ(f )

) ∼=−→Tr(ρ(g)ρ(f )ρ

(g−1)).

Composed with Tr(φg,f,g−1) this gives the desired map ψ(g). Properties (a) and (b) of ψ(g)

follow from Proposition 3.8(b) and (c). �Remark 4.11. By regarding G as a discrete topological space, we can consider the correspon-dence g �→ Tr(ρ(g)) as a sheaf of sets on G. If G is a topological or algebraic group, there arenatural situations when Tr(ρ) is a sheaf on G in the corresponding stronger sense, equivariantunder conjugation, see Section 5.3 below for an example.

Definition 4.12. If ρ is a 2-representation in a k-linear 2-category with finite-dimensional2- Hom(φ,ψ), we define the categorical character of ρ to be the function χρ on pairs of com-muting elements given by the joint trace of ρ(g) and ρ(h):

χρ(g,h) = τ(ρ(g),ρ(h)

) = Trace{ψ(h) : Tr

(ρ(g)

) → Tr(ρ(g)

)}.

Note that

χρ

(s−1gs, s−1hs

) = χρ(g,h). (3)

This can be formulated as follows.

Definition 4.13. Let G be a group and R be a commutative ring. A 2-class function on G withvalues in R is a function χ(g,h) defined on pairs of commuting elements of G and invariantunder simultaneous conjugation, as in (3). The ring of such functions will be denoted 2 Cl(G;R).

Thus the categorical character is a 2-class function.


5. Examples

5.1. 1-dimensional 2-representations

Let k be a field and

c : G × G → k∗

be a 2-cocycle, i.e., it satisfies the identity

c(g1g2, g3)c(g1, g2) = c(g1, g2g3)c(g2, g3).

We then have an action ρ = ρc of G on Vectk . By definition, for g ∈ G the functor ρ(g) : Vectk →Vectk is the identity, while

φg,h : id = ρ(g) ◦ ρ(h) ⇒ ρ(gh) = id

is the multiplication with c(g,h), and φ1 is the multiplication by c(1,1). The cocycle conditionfor c is equivalent to condition (e) of Definition 4.1, while condition (f) follows because

c(1,1g) · c(1, g) = c(1,1) · c(1, g)

implies that

c(1, g) = c(1,1)

and similarly

c(g,1) = c(1,1).

Cohomologous cocycles define equivalent 2-representations, and it is easy to see that H 2(G, k∗)is identified with the set of G-actions on Vectk modulo equivalence. Compare [23].

We now find the categorical character and the 2-character of ρc. First of all, the functor ρc(g)

being the identity,

Tr(ρc(g)

) = k.

Next, the equivariant structure on Tr(ρc) was defined in the proofs of Propositions 3.8(a) and 4.10to be the composition

Tr(ρc(f )

) u−→ Tr(ρc(g)ρc(f )ρc

(g−1)) −→ Tr

(ρc

(gfg−1)).

Here u is induced by the 1-composition with

u : 1Vectk ⇒ BC,

where B = ρc(g) and C = ρc(g−1). In our case,

u = c(g,g−1)−1


(multiplication with a scalar). The second map is induced by

φg,f,g−1 = c(g,f )c(gf,g−1),

see Definition 4.1(e). As a result we have

Proposition 5.1. For any two commuting elements f,g ∈ G, we have

χρc(f, g) = c(g,f )c(gf,g−1)c(g,g−1)−1

c(1,1)−1.

Notice also the following fact which extends Example 4.3.

Proposition 5.2. (Compare [13].) Let ρc be the one-dimensional 2-representation of G onV = Vectk corresponding to c. Then objects of VG are the same as projective representationsof G with central charge c, i.e., pairs (V ,ϕ : G → Aut(V )), where V is a k-vector space, and ϕ

is a map satisfying

ϕ(gh) = ϕ(g)ϕ(h) · c(g,h).

5.2. Representations on 2-vector spaces

2-representations ρc from Section 5.1 can be viewed as acting on the 1-dimensional 2-vectorspace [1]. More precisely, let

1 → k∗ → Gπ→ G → 1

be the central extension corresponding to the cocycle c. For every g ∈ G the set π−1(g) is thek∗-torsor, and therefore

Lg := π−1(g) ∪ {0}

is a 1-dimensional k-vector space. The group structure on G induces isomorphisms

Lg ⊗k Lh → Lgh.

It follows that associating to g ∈ G the 2-matrix ‖Lg‖ of size 1 × 1 gives a 2-representation ofG on [1] ∈ Ob(2- Vectk).

More generally, a 2-representation ρ of G on [n] consists of the following data: for eachg ∈ G, a quasi-invertible 2-matrix ρ(g) = ‖ρ(g)ij‖ of size n × n, with each ρ(g)ij being ak-vector space, plus the data φg,h as in Definition 4.1.

Lemma 5.3. A 2-matrix A = ‖Aij‖ of size n × n is quasi-invertible if and only if there is apermutation σ ∈ Σn such that Aij = 0 for i �= σ(j) and dim(Ai,σ (i)) = 1.

It follows that an n-dimensional 2-representation of G defines a homomorphism G → Σn

plus some cocycle data. This is naturally explained in the context of induced 2-representations,cf. Section 7 below.


Remark 5.4. Lemma 5.3 suggests that the theory becomes richer if one works with generaliza-tions of 2 Vect that have more interesting quasi-invertible 1-morphisms. One such generalizationwas defined in [14].

5.3. Constructible sheaves

Let X be a real analytic manifold acted upon by G. We view each g ∈ G as a map g : X → X.Let V be the category DbConstr(X), see Section 2.4(d). The base field k will be taken to be thefield C of complex numbers, for simplicity. We have then an action ρ of G on V given by

ρ(g)(F) = (g−1)∗

(F)

(inverse image under g−1). As in Examples 2.4(c), (d), it is more practicable to lift this action ona category to an action on an object X of the 2-category RAnC. Namely, for g ∈ G we denote byΓ (g) ⊂ X×X its graph and associate to g the constructible sheaf Kg = CΓ (g), the constant sheafon Γ (g). Note that ρ(g) = FKg

is the functor associated to Kg . It is clear that the correspondenceg �→ Kg gives an action of G on the object X which we denote ρ.

Proposition 5.5. Assume that X is oriented. Then the categorical character of ρ is found asfollows:

Tr•(ρ(g)

) = H •+codimXg (Xg,C

),

where Xg ⊆ X is the fixed point locus of g.

Proof. As in Example 3.6, we denote by i : X → X × X the diagonal embedding and we have

Tr(ρ(g)

) = Tr(FKg) = H

•(X, i!(Kg))

and

i!(Kg) = RΓ Δ(Kg) = CXg

[codim

(Xg

)].

Here Δ = i(X) is the diagonal in X × X. �Assume further that X is compact, and G is a Lie group acting smoothly on X. Then Propo-

sition 5.5 can be sheafified as follows. Let

Y = {(g, x) ∈ G × X: gx = x

}be the “universal” fixed point space. We have the natural embedding and projection

G × Xη←− Y

p−→ G. (4)

Further, the group G acts on the left on G × X, preserving Y , by the formula

g0(g, x) = (g0gg−1, g0x

).
0


Thus η is G-equivariant, and so is p, if we consider the action of G on itself by conjugations.Thus the constructible complex of sheaves

T• = Rp∗CY (5)

on G is conjugation equivariant. It can be seen as a sheaf-theoretical version of the categoricaltrace. Indeed, for any g ∈ G the complex T•

g , the stalk of T• at any g ∈ G has cohomology

Hi(T•

g

) = Hi(Xg,C

) = Tr•−codim(Xg

)(ρ(g)

). (6)

Example 5.6 (Character sheaves). We specialize to the case when G is a complex semisimplealgebraic group and X is the flag variety of G, i.e., the space of all Borel subgroups in G withG-action by conjugation. In this case a class of conjugation equivariant complexes on G wasconstructed by G. Lusztig in the framework of his theory of character sheaves [29]. Lusztig’scomplexes are grouped into “series” labeled by an element w of the Weyl group. We considerhere the “principal” series corresponding to w = 1. Complexes of this series are defined in termsof the diagram (3) and can be interpreted as categorical traces of certain 2-representations of G,via a twisted version of Proposition 5.5 and the equality (6).

To be precise, recall that all Borel subgroups B ⊂ G are conjugate and the normalizer of anyB is B itself. Therefore, the abelianizations B/[B,B] for different B are canonically identifiedwith each other. Equivalently, we can say that they are all identified with a fixed group T (the“abstract” maximal torus, cf. [9, p. 137]). Since in our case

Y = {(g,B) ∈ G × X: g ∈ B

},

we get a projection q : Y → T taking (g,B) to the image of g in the abelianization of B . Givena 1-dimensional local system L on T , the sheaf q∗L on Y is G-equivariant, and hence the con-structible complex

T•(L) = Rp∗q∗L (7)

on G is conjugation invariant. The complex from (5) corresponds to L= CT . Lusztig’s charactersheaves (corresponding to w = 1) are direct summands of T•(L) in the derived category.

This can be interpreted as follows. Let π : Z → X be the basic affine space of G. It is a prin-cipal T -bundle on X isomorphic to the product of the C

∗-bundles corresponding to fundamentalweights, see, e.g., [8, Sect. 2.3.2]. The fibers π−1(B),B ∈ X, are identified with T uniquely upto a group translation in T . Although the local system L on T is not T -equivariant (unless itis trivial), each translation takes it into an isomorphic sheaf. Therefore it makes sense to speakabout local systems on π−1(B) isomorphic to L on T .

Call an L-twisted sheaf on X a sheaf on Z whose restriction on each fiber of π has the formL′ ⊗ V where L′ is a local system isomorphic to L on T , and V is a C-vector space. Let

V(L) = DbLConstr(X)

be the derived category of bounded complexes of L-twisted sheaves on X with constructiblecohomology. We have a natural action ρL of G on V(L), and the stalk of T•(L) at g can berelated to the categorical trace of ρL(g) similarly to (6). As before, to make this precise, we


need to lift ρL to an action ρL by “kernels” and restrict the kernels to the diagonal. Compare [8,Sections 3.3–4].

6. Representations of finite groupoids

6.1. Reminder on semisimplicity

Recall that a groupoid is a category with all morphisms invertible. A groupoid is called finiteif it has finitely many objects and morphisms. As before, we fix a field k of characteristic 0containing all roots of unity. We denote by Vectk the category of finite-dimensional k-vectorspaces.

Definition 6.1. Let G be a finite groupoid. A (finite-dimensional) representation over k of G is afunctor from G to Vectk . A morphism between two G-representations is a natural transformationbetween them. We denote the category of G-representations over k by Rep(G). Object-wisedirect sum and tensor product make Rep(G) into a bimonoidal category, so that its Grothendieckgroup R(G) (with respect to the direct sum) is a ring, the representation ring of G.

Definition 6.2. Let α : H → G be a map of finite groupoids. Then precomposition with α definesa functor

res |α :Rep(G) → Rep(H).

If H is a subgroupoid of G and α is its inclusion we also denote res |α by res |GH .

Definition 6.3. Let G be a finite groupoid. The groupoid algebra k[G] has as underlying k-vector space the vector space with one basis-element eg for each morphism g of G. The algebrastructure is given by

eg · eh ={

egh if g and h are composable,0 else.

The categories of G-representations over k and of k[G]-modules are then equivalent.

Proposition 6.4. If α : H → G is an equivalence of groupoids, then

res |α : Rep(G) → Rep(H)

is an equivalence of categories.

Corollary 6.5. If the groupoids G and H are equivalent, then their groupoid algebras are Moritaequivalent.

Observation 6.6. Every finite groupoid is equivalent to a disjoint union of groups.

Proof. Let G be a finite groupoid. Pick a representative for each isomorphism class of objectsin G. Consider the inclusion of the disjoint union of the automorphism groups of these represen-tatives in G. By construction, this inclusion is fully faithful and essentially surjective, so it is anequivalence of categories. �


Corollary 6.7. The groupoid algebra of a finite groupoid G is semi-simple. Thus there is a uniquedecomposition of representations of G into irreducibles.

Definition 6.8. (a) Let G be a groupoid. The inertia groupoid Λ(G) of G has as objects, theautomorphisms of G (i.e., morphisms u ∈ Mor(G) whose source and target coincide). For twosuch morphism u,v there is one morphism in Λ(G) from u to v for every morphism g of G withv = gug−1.

(b) A class function on a groupoid G is a function defined on isomorphism classes of objectsof Λ(G).

(c) Let ρ be representation of a finite groupoid G. The character of ρ is the k-valued classfunction on G given by

χρ

([g]) = Trace(ρ(g)

).

As before for the case of groups, we denote by Cl(G; k) the ring of class functions on G.

Corollary 6.9. Sending a representation to its character is a ring map R(G) → Cl(G; k). Thismap becomes an isomorphism after tensoring with k.

Proof. The first statement is obvious, the second one follows from Observation 6.6 and Propo-sition 4.7. �6.2. Induced representations of groupoids

Definition 6.10. Let α : H → G be a map of groupoids, and let V be a representation of H .Viewing V as a k[H ]-module, we define the induced G-representation of V by

ind |α(V ) := k[G] ⊗k[H ] V.

We will sometimes write ind |GH for ind |α , if the map is obvious.

Note that ind |α is left adjoint to res |α .Let

α : H → G

be faithful and essentially surjective, and let ρ be a representation of H . By Observation 6.6,we may assume that G and H are disjoint unions of groups. Then a representation of G can bedescribed one group at a time, so we may as well assume that G is a single group and

α : H1 � · · · � Hn → G

is given by a (nonempty) set of injective group maps α1, . . . , αn.In this situation, the induced representation of ρ along α is isomorphic to

ind |α(ρ1, . . . , ρn) :=⊕

ind |αiρi .

Note also that away from the essential image of α, the induced representation along α is zero.


Proposition 6.11. Assume that α is a faithful functor, let x be an object of G and g ∈ HomG(x, x).Then the character of the induced representation evaluated at g is given by the formula

χind(x, g) =∑y∈H0

1

|orbitH (y)|1

|AutH (y)|∑

s∈G1|sx=y

sgs−1∈AutH (y)

χ(y, sgs−1).

Here orbitH (y) is the H -isomorphism class of y, and the second sum is over all morphisms s ofG with source x and target y that conjugate g into a morphism in the image of α|AutH (y).

Proof. Let orbitG(x) denote the G-isomorphism class of x, and let

R = {y1, . . . , yn}

be a system of representatives for the H -isomorphism classes mapping to orbitG(x) under α. Foreach j , pick an sj with sj x = α(yj ). Denote α|AutH (yj ) by αj . We have

χind =n∑

j=1

χind |αj

and

χind |αj(x, g) = χind |αj

(sj x, sj gs−1

j

).

By the classical formula for the character of an induced representation of a group, we have

χind|αj

(sj x, sj gs−1

j

) = 1

|AutH (yj )|∑

t∈AutG(yj )

tsj gs−1j t−1∈AutH (yj )

χ(yj , tsj gs−1

j t−1)

= 1

|AutH (yj )|∑

sx=yj

sgs−1∈AutH (yj )

χ(yj , sgs−1).

The first sum in the proposition is over all objects y of H . If α(y) is not isomorphic to x, thenthe second sum is empty. For the ones isomorphic to x, we are double counting: rather than justhaving one summand for the representative yj , we have the same summand for every element inits H -orbit. �Corollary 6.12. Consider an inclusion of groups H ⊆ G and the induced map of inertiagroupoids α : Λ(H) → Λ(G). Let ρ be a representation of Λ(H). Then the character of theinduced representation ind |αρ evaluated at a pair of commuting elements of G is given by theformula


χ(g1, g2) =∑

h1∈H

1

|[h1]H |1

|CH (h1)|∑

sg1s−1=h1

sg2s−1∈CH (h1)

χ(sg1s

−1, sg2s−1)

= 1

|H |∑

s(g1,g2)s−1∈H×H

χ(sg1s

−1, sg2s−1).

Here [h1]H is the conjugacy class of h1 in H while CH (h1) is the centralizer of h1 in H .

7. Induced 2-representations

7.1. Main definitions

Definition 7.1. Let H ⊆ G be an inclusion of finite groups. Let

ρ : H → Fun(V,V)

be an action of H on a category V . Let ind |GH (V) be the category whose objects are maps

f : G → ObV

together with an isomorphism for every g ∈ G and h ∈ H

ug,h : f (gh) → ρ(h−1

)(f (g)

)satisfying the following two conditions: First, it is required that

ug,1 : f (g) → ρ(1)(f (g)

)coincides with φ−1

1,f (g), see Definition 4.1(d). Second, it is required that for every g ∈ G and everyh1, h2 ∈ H , the diagram

f (gh1h2)ugh1,h2

ug,h1h2

ρ(h−11 )(f (gh2))

ρ(h−12 )ug,h1

ρ((h1h2)−1)(f (g)) ρ(h−1

2 )ρ(h−11 )(f (g))

φh−11 ,h

−12

commutes.A morphism in ind |GH (V) between two systems (f,u = (ug,h)) and (f ′, u′ = (u′

g,h)) is asystem of morphisms f (g) → f ′(g) in V , given got each g ∈ G and commuting with the ug,h

and u′ .
g,h


We define a left action σ of G on ind |GH (V) by

(σ(g1)f

)(g) = f

(g−1

1 g),

(σ(g1)u

)g,h

= ug−1

1 g,h.

Remark 7.2. Consider the category∏

g∈G V whose objects are all maps G → ObV . This cate-gory has a left H -action ξ given by(

ξ(h)f)(g) = ρ(h)

(f (gh)

).

One sees immediately that

ind∣∣GH

(V) =( ∏

g∈G

V)H

is identified with the category of H -equivariant objects in∏

g∈G V .

An explicit description of ind |GHV is given as follows (compare this to the classical definitionof the induced representation as in, e.g., [38]):

Let m be the index of H in G. The underlying category of ind |GHV is then identified with Vm.Such an identification is obtained by picking a system of representatives

R= {r1, . . . rm}of left cosets of H in G and associating to every map f as above the system (f (r1), . . . , f (rm)).

We view ind |GH ρ(g) as m × m matrix whose entries are functors from V to V . Then

(ind

∣∣GH

ρ(g))ij

={

ρ(h) if grj = rih, h ∈ H,

0 else.

Note that in each row and each column, there is exactly one block entry, and that therefore sucha matrix gives a functor from Vm to Vm.

Composition is defined as follows:

(ind

∣∣GH

ρ(g1)) ◦1

(ind

∣∣GH

ρ(g2))ik

={

ρ(h1) ◦1 ρ(h2) if g1rj = rih1 and g2rk = rjh2,

0 else.

In the case that this is not zero,

(ind

∣∣GH

ρ(g1g2))ik

= ρ(h1h2),

since in this case

g1g2rk = rih1h2.

On this block the composition isomorphism is given by the 2-isomorphism

ρ(h1) ◦1 ρ(h2) ⇒ ρ(h1h2).


Similarly, the isomorphism

ind∣∣GH

ρ(1) ⇒ 1Vm

is given by the corresponding map for ρ.

Proposition 7.3. (Compare [34, Ex. 3.4.,Th. 2] and [13, 6.5].) Let G be a finite group, and letV = Vect⊕n

k . Then any 2-representation ρ of G in V is isomorphic to a direct sum

m⊕i=1

ind∣∣GHi

ρωi,

where the Hi are subgroups of G, ωi ∈ H 2(Hi, k∗), and ρωi

is the 1-dimensional 2-representationcorresponding to ωi . Moreover, the system of (Hi,ωi) is determined by the G-action on Vuniquely up to conjugation.

Proof. By Lemma 5.3, ρ defines a homomorphism from G to Σn, i.e., a G-action on the set{1, . . . , n}. Let O1, . . . ,Om be the orbits of this action. It follows that, after renumbering of1, . . . , n, the 2-matrices ρ(g) become block diagonal with blocks of sizes |O1|, . . . , |Om|. Hence

ρ ∼=m⊕

i=1

ρi.

Let Hi be the stabilizer of an element of Oi . For h ∈ Hi , the 2-matrix ρi(h) is diagonal with thesame 1-dimensional vector space Li(h) on the diagonal. We conclude that

ρi∼= ind

∣∣GHi

(ρωi),

where ρωiis the 1-dimensional 2-representation of Hi corresponding to the system (Li(h),

h ∈ Hi). �Remark 7.4. It follows that in the particular case of the proposition, the approach of Ostrik isequivalent to ours. In fact, the category

(ind

∣∣GH

(ρω))G

, H ⊆ G, ω ∈ H 2(H,k∗),is identified, as a Rep(G)-module category, with the category of projective representations of H

with the central charge ω.

7.2. The character of the induced 2-representation

The aim of this subsection is to prove the following theorem.

Theorem 7.5. Let V be k-linear. The categorical trace Tr takes induced 2-representations intoinduced representations of groupoids. That is,


Tr(ind

∣∣GH

ρ) ∼= ind

∣∣Λ(G)

Λ(H)

(Tr(ρ)

),

as representations of Λ(G).

Corollary 7.6. (Compare [21, Thm. D].) Assume that the Tr(ρ(h)) are finite-dimensional. Let χ

denote the 2-character of ρ. The 2-character of the induced representation is given by

χind(g,h) = 1

|H |∑

s−1(g,h)s∈H×H

χ(s−1gs, s−1hs

).

Proof Theorem 7.5. We want to compute

χind := Tr(ind

∣∣GH

ρ)

as representation of

Λ(G) �∐[g]G

CG(g).

Let R be a system of representatives of G/H , and fix g ∈ G. The underlying vector space ofχind(g) is the sum over all r ∈R which produce a diagonal block entry in indρ(g),

χind(g) =⊕

r−1gr∈H

Tr(ρ(r−1gr

))(8)

(compare [38]). We need to determine the action of CG(g) on χind(g). For this purpose, wereplace our system of representatives R in a convenient way: The decomposition

[g]G ∩ H = [h1]H ∪ · · · ∪ [hl]Hinduces a decomposition

{r ∈R

∣∣ r−1gr ∈ H} =

l⋃i=1

Ri ,

with

Ri := {r ∈ R

∣∣ r−1gr ∈ [hi]H}.

We fix i, pick ri ∈ Ri , and write hi := ri−1gri .

Lemma 7.7. We can replace the elements of Ri in such a way that left multiplication with r−1i

maps Ri bijectively into a system of representatives of

CG(hi)/CH (hi).


Proof. If r ∈Ri satisfies

r−1gr = h−1hih,

we replace r by rh−1, which represents the same left coset of G/H as r does. Note that

(rh−1)−1

grh−1 = hi. (9)

We have (r−1i rh−1)−1

hi

(r−1i rh−1) = hi,

therefore r−1i (rh−1) ∈ CG(hi). Assume now that we have replaced Ri in this way. To prove that

left multiplication with r−1i is injective, let r �= r ′ ∈Ri . Then

(r−1i r

)−1r−1i r ′ = r−1r ′

is not in H , and therefore r−1i r ′ and r−1

i r are in different left cosets of CH (hi) in CG(hi). Toprove surjectivity, let g ∈ CG(hi). Write

ri g = rh

with r ∈R and h ∈ H . Then

r−1gr = hg−1ri−1gri gh−1 = hg−1high−1 = hhih

−1.

Therefore, r is in Ri , and it follows from the identity (9) that

r−1gr = hi.

Thus r−1i r = gh is in the same left coset of CH (hi) in CG(hi) as g is. �

Let αi denote the composition

CH (hi) ↪→ CG(hi)→CG(g),

where the second map is a conjugation by r−1i . Recall that as a representation of CG(g),

(ind

∣∣Λ(G)

Λ(H)π

)(g) =

l⊕i=1

ind |αiπ(hi). (10)

Lemma 7.8. As a representation of CG,

χind(g) ∼=l⊕

i=1

ind |αiTr

(ρ(hi)

).


Proof. Let f ∈ CG(g), and let r ∈Ri . Write

f r = rh,

with r ∈R and h ∈ H . We claim that r is also in Ri and that h is in CH (hi). This follows from

r−1gr = hr−1f −1gf rh−1 = hr−1grh−1 = hhih−1 = hi

as in the proof of Lemma 7.7. We are now ready to compute the block entry corresponding to(r, r) of

indρ

(f −1) ◦1 indρ(g) ◦1 indρ(f ):

f r = rh gives(indρ(f )

)rr

= ρ(h),

gr = rhi gives(indρ(g)

)r r

= ρ(hi),

f −1r = rh−1 gives(indρ

(f −1))

rr= ρ

(h−1)

and all other block entries in these rows and columns are zero. Thus(indρ

(f −1) ◦1 indρ(g) ◦1 indρ(f )

)rr

= ρ(h−1) ◦1 ρ(hi) ◦1 ρ(h), (11)

and the 2-morphism from (11) to (indρ(g)

)rr

= ρ(hi)

is the 2-morphism

ρ(h−1) ◦1 ρ(hi) ◦1 ρ(h) ⇒ ρ(hi)

corresponding to h−1hih = hi . This proves that the action of CG(g) on χind(g) decomposes intoactions on ⊕

r∈Ri

ρ(hi).

More precisely, if f r = rh, then f maps the summand corresponding to r to the one correspond-ing to r by

h : ρ(hi) → ρ(hi).

But

f r = rh ⇐⇒ (ri

−1f ri)(

r−1i r

) = (r−1i r

)h,

and the action of

ri−1f ri ∈ CG(hi)


on

ind∣∣CG(hi)

CH (hi )ρ(hi)

is given by

h : r−1i rρ(hi) → r−1

i rρ(hi).

Lemma 7.8 is proved. �This now completes the proof of Theorem 7.5. �

8. Some further questions

8.1. Inertia orbifolds

Recall that a Lie groupoid is a groupoid Γ enriched in the category of C∞-manifolds, i.e., suchthat Ob(Γ ) and Mor(Γ ) are C∞-manifolds and all the structure maps (composition, inverses,units) are smooth. See [30] for more details. An orbifold, cf. [32], is a Lie groupoid G such thatall stabilizer groups

HomΓ (x, x), x ∈ Ob(Γ ),

are finite. The construction of an inertia groupoid Λ(Γ ), see Definition 6.8, can be applied to aLie groupoid (resp. orbifold) Γ and the result is again a Lie groupoid (resp. orbifold).

Example 8.1 (Global quotient groupoids). Let M be a manifold and G be a Lie group actingon M . Then we have a Lie groupoid M//G with

Ob(M//G) = M, HomM//G(x, y) = {g ∈ G: g(x) = y

}.

Thus Mor(M//G) = G × M . If the stabilizer of each x ∈ M is finite, then M//G is an orbifold,known as the global quotient orbifold. The latter condition is automatically satisfied, if G itselfis finite. In this case the inertia orbifold of M//G can be identified as follows:

Λ(M//G) =∐[g]G

Mg//CG(g).

Here the disjoint union is over the conjugacy classes in G, and Mg stands for the g-fixed pointlocus of M .

Recall further that equivariant K-theory of a manifold with a finite group action is a particularcase of a more general concept of orbifold K-theory Korb(Γ ) defined for any orbifold Γ . Thisparticular case corresponds to a global quotient orbifold:

Korb(M//G) ∼= KG(M).


Our 2-character map

Tr : 2 Rep(G) → Rep(Λ(G)

)(12)

should be compared to the orbifold Chern character map defined by Adem-Ruan and interpretedby Moerdijk [32, p. 18] as a map

K•(Γ ) ⊗ C →∏i

H 2i+•(Λ(Γ ),C). (13)

Here Γ is any orbifold whose quotient space (i.e., the space of isomorphism classes of objects)is compact.

This suggests that (12) has a generalization for an arbitrary orbifold Γ as above, yielding atransformation

2Korb(Γ ) → Korb(Λ(Γ )

).

Here 2Korb(Γ ) is a (yet to be defined) orbifold/equivariant version of the 2-vector bundle K-theory of [3]. Recall that the non-orbifold 2K is interpreted as some approximation to ellipticcohomology. Therefore the orbifold version is to be regarded as a geometric version of equivari-ant elliptic cohomology, thus making more precise our point in the introduction. Note that inertiaorbifolds also turn up in the original paper [21], where working at chromatic level n requiresusing n-fold iterated inertia orbifolds Λn(Γ ).

8.2. The Todd genus of Xg

Let G be a finite group acting on a compact d-dimensional complex manifold X. For g ∈ G,the fixed point locus Xg is then a compact complex submanifold. Consider the graded vectorspaces

T(g) := H •(Xg,O),

where O is the sheaf of holomorphic functions on Xg . Clearly, these are conjugation equivariant,i.e., they form a representation of Λ(G). Its character is a 2-class function

χX ∈ 2 Cl(G;Z[ζN ]), χX(g,h) = Trace

{h∗ : T(g) → T(g)

}.

This function takes values in the cyclotomic ring Z[ζN ], where ζN is an N th root of 1 and N isthe order of G.

On the other hand, the Hopkins–Kuhn–Ravenel theory [21] also provides a 2-class functionassociated with the G-manifold X. Let [X] ∈ MU(BG) denote the image of the equivariantcobordism class of X. Fix a prime p, and let E = E2 be the second Morava E-theory at p.Recall that E comes with a canonical natural transformation of cohomology theories

φ : MU∗(−) → E∗(−).

Let now G be finite. In this situation Hopkins, Kuhn and Ravenel constructed a map

α : E∗(BG) → 2 Cl(G;D),


for a certain ring

D = lim−→n

Dn,

where Dn is known as the ring of Drinfeld level pn structures on the formal group E∗(pt):

Dn := E0(B(Z/pn

Z)2)

/(annihilators of nontrivial Euler classes).

The ring Dn can be seen as the second chromatic analog of the cyclotomic ring Z[ζpn] whichcorresponds to level pn structures on the multiplicative group. In fact, a version of the Weilpairing [2] shows that Dn contains Z[ζpn ]. The 2-class function a(y), y ∈ E∗(BG) is defined by

a(y)(g,h) := (g,h)∗n(y) ∈ Dn

where (g,h) is a pair of commuting p-power order elements of G and

(g,h)n : (Z/pnZ

)2 → G, pn = max(ord(g),ord(h)

),

is the homomorphism corresponding to (g,h).

Question 1. Is there a natural 2-representation in some category of sheaves associated to X

whose categorical character is T?

Question 2. What is the relationship between a(φ[X])(g,h) and χX(g,h)?

Acknowledgments

Our inspiration for this project came from conversations with Haynes Miller. We would liketo thank Angelica Osorno, Jim Stasheff and Simon Willerton for their remarks on earlier versionsof this paper. We are grateful to the referee, whose comments greatly helped to make the paperaccessible for a larger audience. We would also like to thank Matthew Ando, Alex Ghitza, andCharles Rezk for many helpful conversations.

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